Theorem tfsconcatfn

Description: The concatenation of two transfinite series is a transfinite series. (Contributed by RP, 22-Feb-2025)

	Ref	Expression
Hypothesis	tfsconcat.op	⊢ + = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( dom 𝑎 +o dom 𝑏 ) ∖ dom 𝑎 ) ∧ ∃ 𝑧 ∈ dom 𝑏 ( 𝑥 = ( dom 𝑎 +o 𝑧 ) ∧ 𝑦 = ( 𝑏 ‘ 𝑧 ) ) ) } ) )
Assertion	tfsconcatfn	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( 𝐴 + 𝐵 ) Fn ( 𝐶 +o 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		tfsconcat.op	⊢ + = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( dom 𝑎 +o dom 𝑏 ) ∖ dom 𝑎 ) ∧ ∃ 𝑧 ∈ dom 𝑏 ( 𝑥 = ( dom 𝑎 +o 𝑧 ) ∧ 𝑦 = ( 𝑏 ‘ 𝑧 ) ) ) } ) )
2		simpll	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → 𝐴 Fn 𝐶 )
3		simplrl	⊢ ( ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) ∧ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) → 𝐶 ∈ On )
4		simplrr	⊢ ( ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) ∧ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) → 𝐷 ∈ On )
5		simpr	⊢ ( ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) ∧ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) → 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) )
6		tfsconcatlem	⊢ ( ( 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) → ∃! 𝑦 ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) )
7	3 4 5 6	syl3anc	⊢ ( ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) ∧ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) → ∃! 𝑦 ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) )
8	7	ralrimiva	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ∀ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∃! 𝑦 ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) )
9		eqid	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) }
10	9	fnopabg	⊢ ( ∀ 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∃! 𝑦 ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) } Fn ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) )
11	8 10	sylib	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) } Fn ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) )
12		disjdif	⊢ ( 𝐶 ∩ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) = ∅
13	12	a1i	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( 𝐶 ∩ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) = ∅ )
14	2 11 13	fnund	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( 𝐴 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) } ) Fn ( 𝐶 ∪ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) )
15	1	tfsconcatun	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( 𝐴 + 𝐵 ) = ( 𝐴 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) } ) )
16		oaword1	⊢ ( ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) → 𝐶 ⊆ ( 𝐶 +o 𝐷 ) )
17		undif	⊢ ( 𝐶 ⊆ ( 𝐶 +o 𝐷 ) ↔ ( 𝐶 ∪ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) = ( 𝐶 +o 𝐷 ) )
18	16 17	sylib	⊢ ( ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) → ( 𝐶 ∪ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) = ( 𝐶 +o 𝐷 ) )
19	18	eqcomd	⊢ ( ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) → ( 𝐶 +o 𝐷 ) = ( 𝐶 ∪ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) )
20	19	adantl	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( 𝐶 +o 𝐷 ) = ( 𝐶 ∪ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) )
21	15 20	fneq12d	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( ( 𝐴 + 𝐵 ) Fn ( 𝐶 +o 𝐷 ) ↔ ( 𝐴 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ∧ ∃ 𝑧 ∈ 𝐷 ( 𝑥 = ( 𝐶 +o 𝑧 ) ∧ 𝑦 = ( 𝐵 ‘ 𝑧 ) ) ) } ) Fn ( 𝐶 ∪ ( ( 𝐶 +o 𝐷 ) ∖ 𝐶 ) ) ) )
22	14 21	mpbird	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( 𝐴 + 𝐵 ) Fn ( 𝐶 +o 𝐷 ) )